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Burau Representation

Gives a Matrix representation ${\hbox{\sf b}}_i$ of a Braid Group in terms of $(n-1)\times (n-1)$ Matrices. A $-t$ always appears in the $(i,i)$ position.

$\displaystyle {\hbox{\sf b}}_1$ $\textstyle =$ $\displaystyle \left[\begin{array}{ccccc}-t & 0 & 0 & \cdots & 0 \\  -1 & 1 & 0 ...
...& \vdots & \vdots & \ddots & \vdots\\  0 & 0 & 1 & \cdots & 1\end{array}\right]$ (1)
$\displaystyle {\hbox{\sf b}}_i$ $\textstyle =$ $\displaystyle \left[\begin{array}{cccccc}1 & \cdots & 0 & 0 & \cdots & 0 \\  \v...
... 0 & 0 & \ddots & \vdots \\  0 & \cdots & 0 & 0 & \cdots & 1 \end{array}\right]$ (2)


\begin{displaymath}
{\hbox{\sf b}}_{n-1} = \left[{\matrix{1 & 0 & \cdots & 0 & 0...
... 0 & \cdots & 0 & -t \cr 0 & 0 & \cdots & 0 & -t \cr}}\right].
\end{displaymath} (3)

Let $\Psi$ be the Matrix Product of Braid Words, then
\begin{displaymath}
{\hbox{det}({\hbox{\sf I}}-\Psi)\over 1+t+\cdots + t^{n-1}} = \Delta_L,
\end{displaymath} (4)

where $\Delta_L$ is the Alexander Polynomial and det is the Determinant.


References

Burau, W. ``Über Zopfgruppen und gleichsinnig verdrilte Verkettungen.'' Abh. Math. Sem. Hanischen Univ. 11, 171-178, 1936.

Jones, V. ``Hecke Algebra Representation of Braid Groups and Link Polynomials.'' Ann. Math. 126, 335-388, 1987.




© 1996-9 Eric W. Weisstein
1999-05-26